Question

find the area of triangle whose sides are 5cm,12cm and 13cm,aiso find its shortest altitude?

Answer 1

for the area of a triangle, you have the formula,
(s(s-a)(s-b)(s-c))^1/2
where s= (a+b+c)/2
but this is aright angled triangle, so you can easily apply the formula area=1/2axb, where a=12, b=5
for altitude, you can imagine it divides the hypotenuse into 2 parts x and 13-x
h,x,5 (h is length of altitude) form a right angled triangle whereas h,13-x,12 form another.
using pythagoras theorm, solve for x.
then find the altitude.
the shortest altitude is the altitude to the hypotenuse.

Answer 2

You can find are of any triangle by HERON'S fromula if all its sides are known.
Here Sides are - 5 cm, 12 cm and 13 cm.
So, s = (a+b+c+)/2 where a,b, and c are sides of triangle
s = (5+12+13)/2 = 30/2 = 15
Area = $\sqrt{s(s-a)(s-b)(s-c)}$
Area = $\sqrt{15(15-5)(15-12)(15-13)} = \sqrt{15(10)(3)(2)} = \sqrt{(150)(6)} = \sqrt{900}$
So, Area = 30 $cm^{2}$

And the smallest height will be on the largest base. So here the largest side is 13 cm.
So by area formula we have
Area = 1/2 *base * height
But area = 30 $cm^{2}$
so, 30 = 1/2 * 13 *height
height = 60/13 = 4.615 cm